Here \({\cal J}_{AB}{\,=\!\scalebox {0.6}{\mathbf {\left(\begin{array}{cc}\mathbf {0}&\mathbf {1}\\\mathbf {1}&\mathbf {0}\end{array}\right)}}}\)  is the \(\mathbf {O}(D,D)\)  invariant metric which, with its inverse, lowers and raises the \(\mathbf {O}(D,D)\)  indices, \(A,B=1,2,\ldots , 2D\) . Remarkably, DFT can be formulated in terms of any generalised metric that satisfies the defining relations (REF ), and may have evolved, at least formally, to an alternative `pure gravity' which string theory seems to predict. It is by now fully equipped with its own Christoffel symbols, scalar/Ricci/Einstein curvatures [1]}, [2]}, and Einstein equations coupled to extra `matter': \(G_{AB}=8\pi G\,T_{AB}\)  [3]}. This single expression unifies the equations of motion of the fundamental variables, \(\lbrace d,{\cal H}_{AB}\rbrace \) . Besides, as solutions to the condition (REF ), all possible classical geometries which DFT is capable of describing have been classified by two non-negative integers, \((n,{\bar{n}})\)  [4]}. Only the type \((0,0)\)  is Riemannian, amounting to the well-known parametrisation (REF ), while others are non-Riemannian in nature: the upper left \({D\times D}\)  block of the generalised metric is degenerate with `nullity' \({n+{\bar{n}}}\) , and thus does not yield an invertible Riemannian metric. For a non-Riemannian geometry to be a consistent string background at the quantum level, it turns out necessary to put \({n={\bar{n}}}\)  in the usual critical dimensions, \({D=10}\)  or 26 [5]}. Non-relativistic string [6]}, [7]}, [8]}, or torsional/stringy Newton–Cartan gravities of recent interest [9]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]}, [16]}
are of the type \((1,1)\)  [17]}, [18]}, [19]}, [20]}, [21]}, [22]}, [23]}, [24]}. The condition \({n={\bar{n}}}\)  further makes the non-Riemannian geometry compatible with type II supersymmetric DFT and superstring [25]}, [18]} (c.f. [27]}): for spin group \(\mathbf {Spin}(t,s){\times \mathbf {Spin}(s,t)}\)  with \({t+s=10}\) , the allowed range of \({n={\bar{n}}}\)  is from zero to \(\min (t,s)\)  [4]}.
